A note on the nonexistence of solutions for prescribed mean curvature equations on a ball

We prove the nonexistence of solutions for a prescribed mean curvature equation {−div(∇u1+|∇u|2)=λ|u|p−1u,x∈BR⊆Rn,u=0,x∈∂BR, when p⩾1p⩾1 and the positive parameter λλ is small. The result extends theorems of Narukawa and Suzuki, and Finn, from the case of n=2,p=1n=2,p=1 to all n⩾2,p⩾1n⩾2,p⩾1. Moreover, our proof is very simple and the result is not limited to positive (and negative) solutions. We also show that a similar result for positive solutions is still true if |u|p−1u|u|p−1u is replaced by the exponential nonlinearity eu−1eu−1.